Diesis
Encyclopedia
In classical music
from Western culture
, a diesis (difference, Greek: "leak or "escape") is either an accidental
(see sharp
), or a comma
type of musical interval, usually defined as the difference between an octave
(in the ratio
2:1) and three justly tuned
major thirds (tuned in the ratio 5:4), equal to 128:125 or about 41.06 cent
s. In 12-tone equal temperament (on a piano for example) three major thirds in a row equal an octave, but three justly-tuned major thirds fall quite a bit narrow of an octave, and the diesis describes the amount by which they are short.
For instance, an octave (2:1) spans from C to C', and three justly tuned major thirds (5:4) span from C to B (namely, from C, to E, to G, to B). The difference between C-C' (2:1) and C-B (125:64) is the diesis (128:125). Notice that this coincides with the interval between B and C', also called a diminished second
.
The 128:125 interval is also known as a lesser diesis, as opposed to a wider comma (648:625) known as greater diesis. As shown in the picture, in the quarter-comma meantone
tuning system (a tuning system in which, by definition, major thirds are justly tuned), the diminished second
coincides with the diesis.
. The diminished second is an interval between pairs of enharmonically equivalent notes; for instance the interval between E and F. As mentioned above, the term diesis most commonly refers to the diminished second in quarter-comma meantone temperament. Less frequently and less strictly, the same term is also used to refer to a diminished second of any size. In third-comma meantone, the diminished second is typically denoted as a greater diesis (see below).
In quarter-comma meantone, since major thirds are justly tuned, the width of the diminished second coincides with the above mentioned value of 128:125. Notice that 128:125 is larger than a unison
(1:1). This means that, for instance, C' is sharper than B. In other tuning systems, the diminished second has different widths, and may be smaller than a unison (e.g. C' may be flatter than B):
In eleventh-comma meantone, the diminished second is within 1/716 (0.0014) of a cent
above unison, so it closely resembles the 1:1 unison ratio of twelve-tone equal temperament
.
The word diesis has also been used to describe a large number of intervals, of varying sizes, but typically around 50 cents. Philolaus
used it to describe the interval now usually called a limma
, that of a justly tuned perfect fourth (4:3) minus two whole tones (9:8), equal to 256:243 or about 90.22 cents. Other theorists have used it for various other intervals.
6:5), which is equal to three syntonic comma
s minus a schisma
, equal to 648:625 or about 62.57 cents (almost one 63.16-cent division in 19 equal temperament
). Being larger, this diesis was termed "greater" while the 128:125 diesis was termed "lesser".
The small diesis is 3125:3072 or approximately 28 cents.
(or slendro
diesis) is an interval
with the ratio of 49:48 , which is the difference between the septimal whole tone
and the septimal minor third
. It is about 35.70 cents wide
The undecimal diesis is equal to 45:44 or about 38.91 cents, closely approximated by 31 equal temperament
's 38.71 cent interval.
Classical music
Classical music is the art music produced in, or rooted in, the traditions of Western liturgical and secular music, encompassing a broad period from roughly the 11th century to present times...
from Western culture
Western culture
Western culture, sometimes equated with Western civilization or European civilization, refers to cultures of European origin and is used very broadly to refer to a heritage of social norms, ethical values, traditional customs, religious beliefs, political systems, and specific artifacts and...
, a diesis (difference, Greek: "leak or "escape") is either an accidental
Accidental (music)
In music, an accidental is a note whose pitch is not a member of a scale or mode indicated by the most recently applied key signature. In musical notation, the symbols used to mark such notes, sharps , flats , and naturals , may also be called accidentals...
(see sharp
Sharp (music)
In music, sharp, dièse , or diesis means higher in pitch and the sharp symbol raises a note by a half tone. Intonation may be flat, sharp, or both, successively or simultaneously...
), or a comma
Comma (music)
In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...
type of musical interval, usually defined as the difference between an octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
(in the ratio
Interval ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2 , 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12, 1.498...
2:1) and three justly tuned
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
major thirds (tuned in the ratio 5:4), equal to 128:125 or about 41.06 cent
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
s. In 12-tone equal temperament (on a piano for example) three major thirds in a row equal an octave, but three justly-tuned major thirds fall quite a bit narrow of an octave, and the diesis describes the amount by which they are short.
For instance, an octave (2:1) spans from C to C', and three justly tuned major thirds (5:4) span from C to B (namely, from C, to E, to G, to B). The difference between C-C' (2:1) and C-B (125:64) is the diesis (128:125). Notice that this coincides with the interval between B and C', also called a diminished second
Diminished second
In modern Western tonal music theory a diminished second is the interval between notes on two adjacent staff positions, or having adjacent note letters, whose alterations cause them, in ordinary equal temperament, to have no pitch difference, such as B and C or B and C...
.
The 128:125 interval is also known as a lesser diesis, as opposed to a wider comma (648:625) known as greater diesis. As shown in the picture, in the quarter-comma meantone
Quarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
tuning system (a tuning system in which, by definition, major thirds are justly tuned), the diminished second
Diminished second
In modern Western tonal music theory a diminished second is the interval between notes on two adjacent staff positions, or having adjacent note letters, whose alterations cause them, in ordinary equal temperament, to have no pitch difference, such as B and C or B and C...
coincides with the diesis.
Alternative definitions
In any tuning system, the deviation of an octave from three major thirds, however large that is, is typically referred to as a diminished secondDiminished second
In modern Western tonal music theory a diminished second is the interval between notes on two adjacent staff positions, or having adjacent note letters, whose alterations cause them, in ordinary equal temperament, to have no pitch difference, such as B and C or B and C...
. The diminished second is an interval between pairs of enharmonically equivalent notes; for instance the interval between E and F. As mentioned above, the term diesis most commonly refers to the diminished second in quarter-comma meantone temperament. Less frequently and less strictly, the same term is also used to refer to a diminished second of any size. In third-comma meantone, the diminished second is typically denoted as a greater diesis (see below).
In quarter-comma meantone, since major thirds are justly tuned, the width of the diminished second coincides with the above mentioned value of 128:125. Notice that 128:125 is larger than a unison
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
(1:1). This means that, for instance, C' is sharper than B. In other tuning systems, the diminished second has different widths, and may be smaller than a unison (e.g. C' may be flatter than B):
- a greater diesis above unison (648:625) for third-comma meantone temperament (see below),
- a diaschismaDiaschismaThe diaschisma is a small musical interval defined as the difference between three octaves and four perfect fifths plus two major thirds . It can be represented by the ratio 2048:2025 and is about 19.5 cents...
above unison (2048:2025) for sixth-comma, - a schismaSchismaIn music, the schisma is the ratio between a Pythagorean comma and a syntonic comma and equals 32805:32768, which is 1.9537 cents...
below unison (32768:32805) for twelfth-comma, and - a Pythagorean commaPythagorean commaIn musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...
below unison (524288:531441) for Pythagorean tuningPythagorean tuningPythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
.
In eleventh-comma meantone, the diminished second is within 1/716 (0.0014) of a cent
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
above unison, so it closely resembles the 1:1 unison ratio of twelve-tone equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
.
The word diesis has also been used to describe a large number of intervals, of varying sizes, but typically around 50 cents. Philolaus
Philolaus
Philolaus was a Greek Pythagorean and Presocratic philosopher. He argued that all matter is composed of limiting and limitless things, and that the universe is determined by numbers. He is credited with originating the theory that the earth was not the center of the universe.-Life:Philolaus is...
used it to describe the interval now usually called a limma
Limma
The word limma or leimma can refer to several different musical intervals, whose only common property is their small size:*The original Pythagorean limma, 256/243, a Pythagorean interval ....
, that of a justly tuned perfect fourth (4:3) minus two whole tones (9:8), equal to 256:243 or about 90.22 cents. Other theorists have used it for various other intervals.
Greater and lesser diesis
Some acoustics texts use the term greater diesis for the difference between an octave and four justly tuned minor thirds (tuned in the ratioInterval ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2 , 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12, 1.498...
6:5), which is equal to three syntonic comma
Syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
s minus a schisma
Schisma
In music, the schisma is the ratio between a Pythagorean comma and a syntonic comma and equals 32805:32768, which is 1.9537 cents...
, equal to 648:625 or about 62.57 cents (almost one 63.16-cent division in 19 equal temperament
19 equal temperament
In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
). Being larger, this diesis was termed "greater" while the 128:125 diesis was termed "lesser".
The small diesis is 3125:3072 or approximately 28 cents.
Septimal and undecimal diesis
The septimal diesisSeptimal diesis
In music, septimal diesis is an interval with the ratio of 49:48 , which is the difference between the septimal whole tone and the septimal minor third. It is about 35.7 cents wide, which is narrower than a quarter-tone but wider than the septimal comma...
(or slendro
Slendro
Slendro is a pentatonic scale, one of the two most common scales used in Indonesian gamelan music, the other being pélog.-Tuning:...
diesis) is an interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
with the ratio of 49:48 , which is the difference between the septimal whole tone
Septimal whole tone
In music, the septimal whole tone, septimal major second, or supermajor second is the musical interval exactly or approximately equal to a 8/7 ratio of frequencies. It is about 231 cents wide in just intonation. Although 24 equal temperament does not match this interval particularly well, its...
and the septimal minor third
Septimal minor third
In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...
. It is about 35.70 cents wide
The undecimal diesis is equal to 45:44 or about 38.91 cents, closely approximated by 31 equal temperament
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
's 38.71 cent interval.